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Superconvergence of spectral collocation and $p$-version methods in one dimensional problems

Author: Zhimin Zhang
Journal: Math. Comp. 74 (2005), 1621-1636
MSC (2000): Primary 65N30, 65N15
Published electronically: March 18, 2005
MathSciNet review: 2164089
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Abstract: Superconvergence phenomenon of the Legendre spectral collocation method and the $p$-version finite element method is discussed under the one dimensional setting. For a class of functions that satisfy a regularity condition (M): $\Vert u^{(k)}\Vert _{L^\infty}\le cM^k$ on a bounded domain, it is demonstrated, both theoretically and numerically, that the optimal convergent rate is supergeometric. Furthermore, at proper Gaussian points or Lobatto points, the rate of convergence may gain one or two orders of the polynomial degree.

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Additional Information

Zhimin Zhang
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Keywords: Spectral collocation method, $p$-version finite element method, exponential rate of convergence, superconvergence
Received by editor(s): April 28, 2004
Received by editor(s) in revised form: July 16, 2004
Published electronically: March 18, 2005
Additional Notes: This work was supported in part by the National Science Foundation grants DMS-0074301 and DMS-0311807
Article copyright: © Copyright 2005 American Mathematical Society